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Properties

Label 2-91-91.80-c1-0-2
Degree $2$
Conductor $91$
Sign $0.887 - 0.459i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0745 + 0.278i)2-s + 1.06i·3-s + (1.66 − 0.958i)4-s + (0.133 − 0.499i)5-s + (−0.296 + 0.0794i)6-s + (−2.03 + 1.69i)7-s + (0.798 + 0.798i)8-s + 1.86·9-s + 0.148·10-s + (−2.70 − 2.70i)11-s + (1.02 + 1.76i)12-s + (−1.12 + 3.42i)13-s + (−0.622 − 0.439i)14-s + (0.532 + 0.142i)15-s + (1.75 − 3.03i)16-s + (−2.26 − 3.91i)17-s + ⋯
L(s)  = 1  + (0.0527 + 0.196i)2-s + 0.615i·3-s + (0.830 − 0.479i)4-s + (0.0598 − 0.223i)5-s + (−0.121 + 0.0324i)6-s + (−0.768 + 0.639i)7-s + (0.282 + 0.282i)8-s + 0.621·9-s + 0.0471·10-s + (−0.816 − 0.816i)11-s + (0.294 + 0.510i)12-s + (−0.312 + 0.950i)13-s + (−0.166 − 0.117i)14-s + (0.137 + 0.0368i)15-s + (0.438 − 0.759i)16-s + (−0.548 − 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.887 - 0.459i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.887 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08015 + 0.263141i\)
\(L(\frac12)\) \(\approx\) \(1.08015 + 0.263141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.03 - 1.69i)T \)
13 \( 1 + (1.12 - 3.42i)T \)
good2 \( 1 + (-0.0745 - 0.278i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 1.06iT - 3T^{2} \)
5 \( 1 + (-0.133 + 0.499i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.70 + 2.70i)T + 11iT^{2} \)
17 \( 1 + (2.26 + 3.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.17 + 2.17i)T + 19iT^{2} \)
23 \( 1 + (7.51 + 4.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.00 + 0.270i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.176 + 0.0474i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.79 - 6.68i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.59 - 2.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.03 - 2.42i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.512 + 0.887i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.503 - 0.134i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 8.53iT - 61T^{2} \)
67 \( 1 + (5.00 - 5.00i)T - 67iT^{2} \)
71 \( 1 + (-1.96 - 7.33i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.20 + 11.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.77 - 6.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.42 + 6.42i)T + 83iT^{2} \)
89 \( 1 + (3.13 + 11.7i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-13.1 + 3.53i)T + (84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.37000444080213740851548259600, −13.14291061357309641002611177083, −11.95984963226928509432166040428, −10.83220468441591278352386664697, −9.894647800691845098374649387160, −8.824891294462928812632971523576, −7.12259284421450564485882611007, −6.03194862804776598648465470939, −4.68030504941030942339436599317, −2.61878977944804227111654767607, 2.23935924684302784037292572617, 3.93656970568412780443056793880, 6.18800939351104234654502559579, 7.23288914303769237847360731004, 7.976082111773505152934390615791, 10.13462965374452986127460556160, 10.56091864684814621427890864945, 12.33422184452527717097269280725, 12.67619194873878244171622029935, 13.67370943263515583481935137177

Graph of the $Z$-function along the critical line